Mechanical and/or chemical removal of material from the
subsurface may generate large subsurface cavities, the destabilisation of
which can lead to ground collapse and the formation of sinkholes. Numerical
simulation of the interaction of cavity growth, host material deformation
and overburden collapse is desirable to better understand the sinkhole
hazard but is a challenging task due to the involved high strains and
material discontinuities. Here, we present 2-D distinct element method
numerical simulations of cavity growth and sinkhole development. Firstly, we
simulate cavity formation by quasi-static, stepwise removal of material in
a single growing zone of an arbitrary geometry and depth. We benchmark this
approach against analytical and boundary element method models of a deep
void space in a linear elastic material. Secondly, we explore the effects of
properties of different uniform materials on cavity stability and sinkhole
development. We perform simulated biaxial tests to calibrate macroscopic
geotechnical parameters of three model materials representative of those in
which sinkholes develop at the Dead Sea shoreline: mud, alluvium and salt.
We show that weak materials do not support large cavities, leading to
gradual sagging or suffusion-style subsidence. Strong materials support
quasi-stable to stable cavities, the overburdens of which may fail suddenly
in a caprock or bedrock collapse style. Thirdly, we examine the consequences
of layered arrangements of weak and strong materials. We find that these are
more susceptible to sinkhole collapse than uniform materials not only due to
a lower integrated strength of the overburden but also due to an inhibition
of stabilising stress arching. Finally, we compare our model sinkhole
geometries to observations at the Ghor Al-Haditha sinkhole site in Jordan.
Sinkhole depth
Sinkholes are enclosed surface depressions in sediments and rocks. They commonly result from subsidence of overburden into void space that is generated through the physical–chemical removal of material in the underground. In the final stage of a sinkhole process, a sudden collapse of the overburden may occur (Gutiérrez et al., 2014; Waltham et al., 2005). Removal of material and void formation in the underground is usually related to hydraulic flow and to associated dissolution, physical erosion of material or both. Subsidence may occur continuously over a long time depending on the flow conditions and material properties (Goldscheider and Drew, 2007; Parise and Gunn, 2007; Waltham et al., 2005). Depending on the properties of the overburden (cover or caprock) and the evolution stages, different sinkhole morphologies can be described. Typical end-members can be defined (Fig. 1; see Gutiérrez et al., 2008, 2014).
Conceptual models of sinkhole formation.
The Dead Sea is a hypersaline terminal lake and is one of the world's most active areas of sinkhole development. More than 6000 sinkholes have formed there at an increasing rate over the last 35 years (Abelson et al., 2017). Previous studies relate the sinkhole formation at the Dead Sea to the regression of the lake, which has been ongoing since the 1960s, and to the consequent invasion of evaporite-rich sedimentary deposits around the Dead Sea by relatively fresh groundwater. Evaporitic minerals in the sediments are susceptible to dissolution, while the non-evaporitic sedimentary materials are weak (poorly consolidated or unconsolidated) and can easily be physically eroded by subsurface flow (“piping”). Some studies have highlighted the role of subrosion, i.e. both mechanical and chemical (leaching) erosion of the subsurface (Wadas et al., 2016), in the development of sinkholes (Al-Halbouni et al., 2017; Arkin and Gilat, 2000; Polom et al., 2018), while others have focussed on the role of dissolution only in generating large cavity development in a relatively shallow but thick salt layer (Ezersky and Frumkin, 2013; Taqieddin et al., 2000; Yechieli et al., 2006).
In this paper, we draw upon observations from the sinkhole site of Ghor
Al-Haditha (31
Sinkhole examples from the eastern shoreline of the Dead Sea.
Sinkholes form in the three “end-member” near-surface materials at the Ghor Al-Haditha sinkhole site (Fig. 2): (1) unconsolidated to semi-consolidated lacustrine clayey carbonates (“mud”) with interleaved thin evaporite layers; (2) unconsolidated to semi-consolidated alluvial sand–gravel sediments; and (3) rock salt (mainly halite) with interleaved thin mud layers. The main morphological distinction is that narrower and deeper sinkholes occur in the “alluvium” and in the “salt” (Fig. 3b, c), whereas wider and shallower sinkholes occur in the “mud” (Fig. 3a). Many sinkholes in the alluvium and especially in the salt have overhanging sides and/or large marginal blocks and deep (up to several metres) concentric ground cracks. The alluvium and the salt can sustain metre-scale or multi-metre cavities associated with sinkhole development (Al-Halbouni et al., 2017; Closson and Abou Karaki, 2009; Yechieli et al., 2006). The mud sinkholes commonly contain a wide peripheral zone of back-rotated blocks delimited by small faults that down-throw towards the centre. Ground cracks are commonly also well developed around sinkholes in the mud but are not as deep (up to a few tens of centimetres) as in the other materials.
Representative morphological data from single sinkholes at the
eastern shoreline of the Dead Sea:
The numerical simulation of sinkhole development is of interest to understand better the processes of sinkhole formation and the related hazard. Continuum-mechanics approaches (Carranza-Torres et al., 2016; Fazio et al., 2017; Fuenkajorn and Archeeploha, 2010; Parise and Lollino, 2011; Rawal et al., 2016; Salmi et al., 2017) have generally defined a single cavity in an elastic or elastoplastic half space and assessed the static threshold strength of the overburden to predict mechanical failure. This approach is possibly suitable for assessing the factor of safety of an individual, fully developed cave and for deriving a relation between measured surface subsidence and cavern configuration. However, the geometries of voids involved in sinkhole development are often non-singular, irregular and distributed on lots of scales (Abelson et al., 2017; Al-Halbouni et al., 2017; Ezersky et al., 2017; Gutiérrez et al., 2016; Parise et al., 2018; Yizhaq et al., 2017). Alternatively, continuum-based corrosion models have addressed the rock dissolution and void growth in a hydrogeological framework (Kaufmann and Romanov, 2016; Shalev and Lyakhovsky, 2012). This approach has the advantage of accounting for geometrically complex or stochastic void development and the role(s) of material heterogeneity, but it does not account for effects of overburden instability. Both of these past continuum-based approaches have neglected the mechanical consequences of void growth and the explicit simulation of sinkhole collapse.
Distinct element method (DEM) modelling is increasingly used in geoscience for numerical simulation of large-strain and discontinuous rock deformation (Cundall and Strack, 1979; Potyondy and Cundall, 2004). The main advantage of the DEM is its ability to simulate rock samples or rock masses as an assemblage of discrete particles or blocks, which can undergo large displacements and rotations. The method uses a so-called soft contact approach where the particles are rigid but allowed to overlap at contact points. Based on updated particle positions, the contacts between particles are automatically detected or deleted during the simulation. Based on the relative displacement and velocity of the particles in contact, interaction laws are used to update the forces and moments transmitted through the contacts. The resultant forces and moments that accumulated on each particle are subsequently used to solve Newton's second law of motion and to update the particle's position and velocity. Elastoplastic bonds of finite strength can be accounted for in the interaction law and enable a quasi-continuum behaviour at assembly scale, which can evolve to highly discontinuous deformation as bonds between particles break and damage develops. In this way, the DEM can overcome limitations of continuum-based numerical simulation of large and highly localised strains in discontinuous media (Jing and Stephansson, 2007). Using the DEM, recent advances have been made in, for example, rock mechanics (Schöpfer et al., 2009), slope stability and mass movements (Thompson et al., 2010), mine or tunnel stability (Bonilla-Sierra et al., 2012), synthetic rock mass modelling (Ivars et al., 2011), fracture growth (Schöpfer et al., 2016), hydrofracture and caldera subsidence analysis (Holohan et al., 2011, 2015, 2017).
For modelling sinkhole collapse, Baryakh et al. (2008, 2009) used the DEM to conduct simple stability tests and mechanical analyses for a single, instantaneously generated cavity of varying geometry, depth and overburden mechanical properties. Other studies have adopted a similar approach but also included discrete fracture networks (DFNs) that represent predefined or empirically determined discontinuities (joints, faults) within rock masses (Hatzor et al., 2010). DEM coupled with finite element modelling (FEM) has been used for simulating mechanical failure above a large salt cavity (Mercerat, 2007), but the DEM part was limited to a single rock layer within the overburden. Again, the main shortcoming of these earlier DEM-based studies is that cavity growth and related mechanical development were not explicitly simulated.
This paper reports the first two-dimensional DEM simulations of sinkhole formation that explicitly simulate both void growth and overburden collapse. In part, the approach of void growth adopted here is similar to that in a recent work on mine caving (Sainsbury, 2012). Our study builds upon the previous works of Caudron et al. (2006) and Baryakh et al. (2008, 2009) but goes further in calibration of geomechanical behaviour, in complexity of process and in application to natural sinkholes. As in many previous studies, we focus here on the creation, growth and instability of a single void leading to a single sinkhole. We use the general-purpose commercial DEM Particle Flow Code in two dimensions (PFC2D) software, developed by Itasca Consulting Group Inc. (Potyondy, 2014b). Further details of the DEM, as implemented in PFC2D, are covered in Appendix A. Note that, in accordance with PFC2D convention, compressive stress is taken as negative throughout this paper.
Regarding the structure of this paper, we begin by summarising tests on model resolution, model dimensions and void creation procedures, and we show results of benchmarking to continuum-based solutions for displacement around a void. We then show the results of calibration tests that were used to tune the bulk geomechanical behaviours of the DEM particle assemblies. Following this, we analyse the evolution stages of model void growth and sinkhole collapse for uniform and layered materials. We then compare the morphological parameters at the Ghor Al-Haditha survey site to those predicted by our models. In the final part, an outlook to future improvements and applications is given.
In this section, we report on convergence and benchmarking tests for the DEM model as pertained to cavity generation. To this end, we firstly simulate a material that behaves elastically by using bond strength and cohesion values at the upper limit of realistic rock strengths (see Table 1). We also report on the material parameter calibration by simulated biaxial compression and tension tests applied to the numerical materials mimicking those common in the Dead Sea region. Finally, we summarise the final procedure for cavity growth that is based on these tests but implemented under conditions in which the DEM model material is weak enough to fail and lead to sinkhole formation.
Setups for model benchmarking, calibration and sinkhole
simulation. Panel
We tested model sensitivity to resolution, dimension and void installation method. In a first test, different void space installation methods were compared in terms of computation time. For this, surface particle displacement was tracked above a cavity of 5 m radius placed at 35 m depth (Fig. 4a), an assumed realistic subrosion zone depth and dimension. Two methods utilised a particle deletion scheme, while two other methods were based on particle radii reduction. No substantial differences in the vertical and horizontal surface displacements were observable; i.e. the methods did not affect the outcome of the elastic solution, but the particle deletion scheme was 1 order of magnitude faster than radii reduction. Hence, the particle deletion scheme was chosen as appropriate for the following tests and the sinkhole models. More details on the results and the set of investigated parameters can be found in Appendix B1.
In a second test, model width, height and particle radii were varied to
determine the optimal model dimensions for the problem of a void space in
the subsurface. The void installation method based on instantaneous particle
deletion was applied. The final results indicates that symmetric boundaries
of
We performed a benchmarking of surface displacements in the DEM cavity development models with displacements derived from different continuum-based approaches. Cavity depth and size, model dimensions (Fig. 4a) and the bulk elastic parameters of the DEM material in Table 1 serve as input parameters for two analytical solutions and a boundary element (BEM) numerical model.
Bulk properties of the particle assemblies used in the benchmarking of DEM cavity formation models vs. analytical solutions and BEM.
The analytical solutions are for a circular cavity in a gravity loaded,
infinite, linear elastic full/half space under plane strain conditions
(Kirsch, 1898; Verruijt and Booker, 2009). The Kirsch
solution is a classical full-space solution for simple excavation shapes
but does not include the free-surface effect; mathematical details are
provided by Brady and Brown (2006). The Mindlin (1940) solution is for
stresses around tunnels and includes free-surface effects; mathematical
details are given by Verruijt and Booker (2009). The input values for the Mindlin
analytical solution are
The BEM model is based on a code by
Nikkhoo and Walter (2015) and simulates the surface displacements
along a cross-section above a 3-D cylindrical void space. The void space is
simulated as a traction-free, horizontal, north–south-oriented cylinder of 200 m
total length. The cylinder's centroid is located exactly beneath the origin;
a hydrostatic remote stress is applied equal to the gravitational stress
The DEM model displacements
Results of benchmarking of the DEM cavity model against
continuum-based cavity models: analytical solutions and modelled
displacement curves for model dimensions of
Based on the above-described tests for model resolution, dimensions and
cavity generation, a generalised setup for cavity growth with attendant
fracturing and sinkhole collapse is presented in
Fig. 4c. This setup comprised a
A key geometric parameter in subsidence studies is the ratio of overburden
thickness (
The geotechnical parameters of rocks and soils commonly cover a wide range, as they depend strongly on detailed mineral composition, grain sizes, external stress conditions, fluid saturation and stress histories; cf., e.g. Brady and Brown (2006) and Jaeger et al. (2007). Here, we consider geotechnical parameters for the three main material types involved in sinkhole formation at the Dead Sea region: (1) lacustrine clayey carbonates and evaporites, (2) alluvial sands and gravels and (3) pure rock salt (halite) (Table 2).
Estimated geomechanical properties of main materials in sinkhole-affected areas at the Dead Sea. References: Ezersky et al. (2017); Ezersky and Livne, (2013); Frydman et al. (2008, 2014); Hoek (2007); Khoury (2002); Manger (1963); Polom et al. (2018); Zhu (2010).
For lacustrine mud, friction angle, cohesion, porosity and density parameters from laboratory tests are used (Ezersky et al., 2013, 2017; Frydman et al., 2008, 2014). For the alluvial sediments, upper limits are given by nearby field investigations in firm sandstone rocks (El-Naqa, 2001) and also by published values for medium-grained Quaternary sand–gravel (Carter, 1983; Manger, 1963; Taqieddin et al., 2000). The bulk modulus of alluvial sand–gravel and lacustrine clays were estimated using Poisson's ratio values from the literature (Zhu, 2010) and shear-wave velocities from recent field measurements (Polom et al., 2018), where the latter were reduced by a factor of 1.5 to account for drained conditions.
Elastic parameters and strength values of the field materials have been
estimated by using tables from Brown (1981) and Hoek (2007) and by
classifying the clayey mud as grade R0 in terms of intact rock consistency
and the alluvial sediments as grade R0–R1. The Holocene salt rock of the
Dead Sea is considered weaker than typical halite rock salt
(Frydman et al., 2008,
2014) and has been classified as grade R1. The cohesion value of salt is
strongly depth dependent and has been determined by using depth-normalised
results derived from triaxial tests (Frydman
et al., 2014) via
Bulk material parameters are determined by simulated, biaxial compression
and tension tests similar to those described by
Khanal and Schubert (2005) and Schöpfer et al. (2007) (see Fig. 4b). We generated material “samples” with
dimensions of
Particle, contact and bond properties for DEM sinkhole collapse models. Note the geology convention.
The sandy gravel and salt materials show brittle failure behaviour (i.e. a sharp post-peak stress drop) at low confining pressures, which changes to brittle–ductile behaviour for larger confining pressures (Fig. 6). Ductile is defined as the state of deformation without significant loss of strength, and the transition to this behaviour is the brittle–ductile transition (Byerlee, 1968). The salty mud material shows a brittle–ductile transition for all tested confining pressures or, more precisely, a brittle-to-cataclastic-flow transition to distinguish it from the brittle-to-crystal-plastic transition (Schöpfer et al., 2013).
Differential stress vs. vertical strain plots for simulated
compression tests. Confining pressures of
Plots of the peak stress data for each confining pressure are used to estimate the bulk strength parameters according to the widely applied Mohr–Coulomb and Hoek–Brown failure criteria (Hoek, 2007; Hoek et al., 2002; Hoek and Brown, 1997) (Fig. 7). The Mohr–Coulomb failure envelopes for the compression tests are shown in Fig. 7a. If tension test results are included, a highly non-linear behaviour of the material is recorded, so that a Mohr–Coulomb envelope is partly not appropriate anymore. Consequently, a non-linear Hoek–Brown envelope is included in Fig. 7b, although it only fits well to all the data for the lacustrine mud material and to low-confining pressure data for the other materials.
The slopes of the elastic parts of the stress–strain curves are used to
estimate the bulk elasticity parameters. Figure 7c shows that Young's modulus,
Bulk failure envelopes and elasticity parameters of the simulated
Dead Sea materials derived from simulated laboratory tests.
Bulk material properties of the three investigated Dead Sea materials as derived by simulated rock tests and measurement circles. All values refer to unconfined conditions (i.e. at or close to the surface). Mohr–Coulomb and Hoek–Brown results are based on compression and tension tests on 10 different particle assemblies for each material.
This calibration shows that the model materials mimic the mechanical responses of the natural materials, and it builds the basis for the analysis of the specific sinkhole formation problem at the Dead Sea, as presented in the following section.
We simulated the effect of continuous material removal from a semi-elliptical subrosion zone at 20, 30 or 40 m depth below the initial surface for all three end-member Dead Sea materials. For brevity, we here report on the evolution of the models with subrosion at 30 m depth only; for the detailed evolution of all simulated configurations, see the electronic Appendix.
As shown in Figs. 8 and 9, the evolution of cavity development strongly depends on the mechanical interaction with the surrounding material. The mud is geomechanically the weakest end-member, and even the initial small cavity is not supported by it; the cavity collapses almost instantly after it is generated. Consequently, a cavity of large size (metre scale) never develops in the mud. As material is progressively removed from the subrosion zone, material from around and above the removal zone subsides gradually toward it. A column of subsiding material develops that is partly fault-bound and is characterised internally by downsagging of the overburden layering. This column grows upward until intersecting the surface, where a sag-like sinkhole forms. With further subrosion, the sinkhole grows deeper and wider as areas marginal to the subsiding column slump inwards.
In contrast, the alluvium is strong enough to sustain the cavity as it grows. The growing cavity interacts mechanically with the surrounding material, as sections of the cavity roof and walls collapse into it. Eventually, the overburden above the cavity fails abruptly, and the cavity is closed by the collapse of the overburden into it. The overburden collapse is also usually partly fault-bound with downsagging or with a more complex internal structure. The resultant model sinkhole margins are characterised initially by large and deep (metre-scale) opening mode fractures (ground cracks), inward-tilted blocks and in part by overhanging sides. With further subsidence, the inward-tilted blocks and overhanging sides tend to slump into the sinkhole's centre. The salt is the strongest end-member geomechanically, and so large stable cavities can develop within it – essentially unaffected by deformation of the surrounding material – until only a thin “bridge” of overburden is left.
The mechanical differences in the structural development are highlighted in Fig. 9. For the low-strength mud, stress arching, which tends to stabilise the overburden, is weakly developed around the material removal zone and within the overburden. Stress arching is well developed around and above the cavity in the alluvium, although the absolute values of shear stress are high on the cavity's lateral walls, suggesting that these areas are most susceptible to failure. The stress arch is disrupted upon final failure of the overburden and formation of a sinkhole. In the strong salt, stress arching is best developed and persists even after the thin “bridge” of the remaining overburden fails.
Evolution of DEM model cavity growth and sinkhole collapse in
end-member Dead Sea materials. Shown here are selected stages in the
development of cavity/sinkhole in salty mud
Evolution of maximum shear stress during cavity growth and
sinkhole collapse in end-member Dead Sea materials. Shown here are selected
stages in the development of cavity/sinkhole in salty mud
We also simulated the effect of continuous material removal from a semi-elliptical subrosion zone at 20, 30 or 40 m depth below the initial surface for layered combinations of the end-member Dead Sea materials. The models comprise a layer of either alluvial sandy gravel or rock salt (0–13 m depth) overlying a lacustrine mud layer (13–40 m depth), followed by the alluvium/salt as a basement, respectively. For brevity, we again report on the evolution of the models with subrosion at a depth of 30 m only (Figs. 10 and 11); for the detailed evolution of all simulated configurations, see the Supplement.
Evolution of DEM model cavity growth and sinkhole collapse in
layered configurations of the end-member Dead Sea materials. Shown here are
selected stages in the development of cavity/sinkhole in salty mud overlain
by alluvium
Evolution of maximum shear stress during cavity growth and
sinkhole collapse in layered configurations of the end-member Dead Sea
materials. Shown here are selected stages in the development of
cavity/sinkhole in salty mud overlain by alluvium
In general, for layered materials with mud as the subrosion-affected
interlayer, the ground tends to fail clearly earlier than for the uniform
materials. The mud cannot sustain large cavities and hence fails immediately
upon material removal, and the upper mud layers bend. This leads
consequently to the development of a cone-shaped underground collapse zone.
In alluvium on lacustrine mud, a small subsidence may be noted before
collapse and cracks appear even at a certain distance from the main area.
Note also the development of ephemeral cavities at the interface with the
mud and/or within the alluvium or salt top layers, as deformation migrates
upward toward the surface. After the collapse, large and small rotated
blocks slump towards the centre, and opening cracks grow downwards to a
depth of 12 m around the collapse zone. These blocks define the base of the
formed sinkholes. Although salt has double the strength of alluvium, the
shapes of the sinkhole for these multilayer models do not differ much, but a
small tendency to more overhanging sides is observed. For the condition of
lacustrine mud on rock salt, the salt layer sustains large cavity formation,
but as soon as the void space reaches
As shown in Fig. 11, the mechanical effect of the weak mud layer is to inhibit the development of stable stress arching in the overburden. Where the weaker layer lies below the stronger layer, the development of a collapse zone is indicated as a zone of low stresses, around and above which a stress arching is weakly developed. Effectively, this subsurface collapse zone is mechanically similar to a large cavity. The lack of support from the weak layer concentrates stress in the stronger layer (note the high magnitude of shear stress there), pushing the strong layer toward failure. Where the weak layer overlies the strong layer, the stress arch is well developed until the cavity growth nears the weaker layer. The weaker layer cannot sustain the stress arch, and so the overburden collapses.
As shown in Fig. 12, the variation of depth of
the subrosion zone changes the morphology of the sinkholes. For more details,
refer to the electronic Appendix. The removed material in the subrosion zone
is assigned a removed “volume”
Sinkhole end-members in dependency of the depth of the subrosion zone for all material combinations investigated in this study. The maximum shear strain is used to visualise the collapse zone.
In lacustrine salty mud, for all subrosion depths, the sinkhole collapse is
gradual with continuous subsidence. The deeper the subrosion zone, the lower
the vertical displacement at the surface, and a greater amount of material
needs to be removed before an effect is visible at the surface (
In the homogeneous alluvium models, the sinkhole collapse process varies
between sudden (shallow material removal zone) and partly gradual (deep
zone). For a shallow subrosion zone, the collapse occurs relatively late at
a removed material volume of
In homogeneous rock salt models, for all subrosion depths, the sinkhole
collapse is sudden and occurs after large amounts of material are removed.
The sinkholes that form are in all cases
For the multilayer model alluvium on mud with alluvial basement, the
collapse in all cases happens earlier than in pure alluvial material and is
sudden. For a shallow subrosion zone, the sinkhole forms at
Similar features are observed for the multilayer model salt on mud; the
collapse in all cases happens earlier than in pure salt material and is
sudden. The removed volume before collapse is similar to results from
alluvium on mud; namely, for a shallow subrosion zone the sinkhole forms at
Figure 13a shows the estimated
Critical thickness to diameter ratios for modelled sinkhole collapse onsets. The error is based on the mean between different particle assemblies for each setting.
Parameters at the onset of collapse.
In Fig. 14, we compare the topographic profiles of sinkholes derived from photogrammetric studies at Ghor Al-Haditha (see Sect. 1, Fig. 3) with our results from DEM sinkhole modelling. In Fig. 14a, we show the simulated sinkhole morphologies for different evolution stages for a subrosion zone with intermediate depth (30 m). To facilitate the comparison, the topographic profiles derived by photogrammetry have been normalised and the axes have been adjusted to the same dimensions as for the models (Fig. 14b). An impressive similarity can be found for these sinkhole end-members both in terms of lateral extent and subsidence amplitude: (1) the mud sinkhole in the field appears to be of an early-stage sinkhole but with a larger extension laterally; (2) the alluvium sinkhole shape is remarkably similar to the late-stage (evolved) modelled sinkholes both laterally and vertically; (3) the salt sinkhole is comparable to the respective simulation result for an early-stage salt sinkhole.
Topographic cross-sections of sinkholes in different cover
materials.
These findings are essentially confirmed by knowledge about the rather
recent development of the sinkholes selected in the mud and salt flats and
the older, more evolved sinkholes in the alluvial fan of Ghor Al-Haditha
(Al-Halbouni et
al., 2017). Our models, which are based on realistic material parameter
estimation, hence reproduce the topographic features of the sinkholes
successfully in the field site. This result is even better reflected in the
De
In Fig. 15, we compare the sinkhole depth
In simulations with uniform mud material, the fit to the De
Sinkhole depth
Depth
Baryakh et al. (2008, 2009) used the DEM to investigate the
effect of depth, geometry and mechanical properties on the collapsed state
in karst. Their approach is to some extent similar to ours; however,
essentially only the position of a rectangular or an arched cavity was
varied for different uncalibrated materials. In contrast, our numerical
simulations allow for a mechanical interaction of a slowly growing void
space with the surrounding rock and provide calibrated bulk rock parameters.
Consequently, the material removal either creates a cavity or not, leading
to variably shaped subsurface collapse zones, details of which are
elaborated on later. Hatzor et al. (2010) used jointed blocky rock mass
(DFN) modelling to define stability criteria (
In summary, earlier studies lack a detailed calibration of the model strength parameters to field and laboratory estimates, and quantitative comparisons of model results with measured data are limited or absent. Our study hence fills this important gap and explicitly simulates cavity growth and related sinkhole development and therefore provides a significant advance in this field.
Our tests and model benchmarking provide several new insights for undertaking the simulation of karstic void development and sinkhole collapse under gravity with the DEM. As expected, there is a strong sensitivity of model results (displacement) not only to parameters such as model dimensions and resolution but also to model shape, with the best results attained for relatively high-resolution and equidimensional model setups. Our tests also show that the method of cavity generation has only a minor impact on the surface displacement pattern. Cavity generation by particle deletion differs from generation by particle radius reduction mainly in the much longer model runtime for the latter. This is reasonable given the elastic and quasi-static conditions of the DEM test models. By such tests, we infer that the models with non-elastic deformation (i.e. cavity wall failure and sinkhole collapse) are also insensitive to cavity generation method as long as they are run under quasi-static conditions, as was the case in our study.
In the benchmarking tests, the DEM surface displacements for a circular
cavity in a gravitationally loaded elastic material closely resemble those
predicted by the BEM model and the Kirsch solution both in the far and
near fields of the subsidence centre (see Fig. 5).
A perfect match is not expected, despite our efforts to compare
like for like, having in mind the intrinsic differences between these models
in terms of material properties and boundary conditions. The Kirsch results
nonetheless provide the best match to the DEM results for both vertical
displacement and displacement differences. Overall, the DEM and Kirsch
curves fit in the near field and behave similarly and realistically
(tendency to zero) in the far field. The BEM models offer a plane-strain
solution for a hydrostatic remote stress, while the two-dimensional DEM
model does not consider out-of-plane stress–strain and additionally has
before cavity creation a horizontal to vertical stress ratio
The manner of cavity growth and its timing relative to collapse are, of course, simplified approximations to complex processes of dissolution and mechanical erosion of the subsurface as they occur in nature. The model cavity grows by instantaneous and repeated material removal of the same volume within a domain of simplified shape. In reality, cavity growth may occur on extremely long to relatively short timescales, depending on the nature of the materials (e.g. limestone vs. salt) and hydrogeological conditions (e.g. porous flow, conduit flow, dripping, flash floods). The cycling to quasi-static equilibrium during each model growth increment ensures, however, that cavity growth rate is smaller than or equal to collapse rate, as expected in nature. An improvement will be to adjust the cavity area growth function to follow typical dissolution laws (cf. Dreybrodt and Kaufmann, 2007; Kaufmann and Romanov, 2016) and thus to develop more complex and realistic cavity geometries.
The outcomes of the simulated compression and tension tests
(Table 4) closely agree with literature values and
estimations from geotechnical studies and seismic velocity measurements
(Table 2), in terms of UCS ranges, bulk densities,
Young's modulus and Poisson ratios. The friction angles of the simulated
sand–gravel and rock salt materials are slightly lower than the desired
values but fit well in the case of the low-strength lacustrine clay material.
Low-friction angles are typical for bonded particle models (cf., e.g.
Schöpfer et al., 2017), because both sliding and rotation
of particles accommodate bulk deformation; with the contact model used in
the present study, the latter cannot be inhibited even with large friction
coefficients. It is well known from other DEM studies that UCS
The Mohr–Coulomb and Hoek–Brown failure envelopes (Fig. 7) fitted to the calibration data serve as guides to the material behaviour. These envelopes were chosen as they are widely used in geomechanics, and so the overall behaviour of the model materials is readily assessed from them. In detail, however, neither envelope provides a perfect fit to the calibration results. This may be a consequence of the timing of confinement of the particle assembly, which is here done before installing the parallel bonds. Our results indicate that this may lead to some stress-path-dependent behaviour that is more complex than can be represented fully by either Mohr–Coulomb or Hoek–Brown envelopes. A thorough exploration of such complexities is well beyond the scope of this paper, but it could be subject of future work.
It is well known that the relationship between field-scale rock parameters and those determined at the laboratory sample scale depends strongly on the degree of fracturing or alteration of the rock mass (Schultz, 1996). Given that the materials we studied are of rather low strength and are weakly consolidated materials (in contrast to hard karst rock in which sinkholes often form), we neglected the effect of pre-existing weaknesses (e.g. tectonic fractures). We hence adopted literature values for salt and mud derived from laboratory-scale measurements. A poorly understood effect in the Dead Sea materials is, however, the influence of water content which may lead to time-dependent geomechanical behaviours (see Shalev and Lyakhovsky, 2012) that is not accounted for in our models. In principle, however, the modelling scheme we developed could be adapted to account for time-dependent (e.g. viscoelastic) material behaviour.
The DEM models of sinkhole collapse show a wide range of structural or morphological features that are found at natural sinkholes, and they highlight how these features reflect the mechanical properties of the material in which the sinkholes form. Similar near-surface structural features are found at volcanic collapse calderas and pit craters, and similar explanation in terms of mechanical properties of the near-surface materials have been proposed (Holohan et al., 2011; Poppe et al., 2015).
In relatively weak materials (here the simulated “mud”), the near-surface strain is distributed across many small fractures, such that there is no sharp margin to the sinkhole. Subsidence at the surface develops gradually before the collapse develops (if at all) and the material's response is brittle–ductile. The sinkhole also widens gradually as it deepens. Overall, the sinkhole formation process is similar to classic “cover sagging” or “cover collapse” with partial suffusion (cf. Gutiérrez et al., 2008).
In relatively strong materials (here the simulated “alluvial sand–gravel” and “salt”), the strain is localised on fewer but larger fractures that develop as faults (shear fractures) and/or deep cracks (opening-mode fractures). Structures like compression ridges form in the centre of the subsidence area. Sinkhole margins in such materials are consequently sharp, steep and, at least initially, overhanging. Any subsidence before collapse is slight, although this depends partly on material rigidity (i.e. modulus); the material's response is brittle. The sinkhole also widens as it deepens but in more of a stepwise manner as new marginal fractures form and delimit marginal blocks. Overall, the collapse style is similar to classic “caprock collapse” or “bedrock collapse” (see Gutiérrez et al., 2008). In extremely strong materials, there may be little or no collapse at all – in the limit, the hole may result simply from the intersection of an essentially stable, growing cavity with the ground surface.
The stability of cavities in the DEM models is clearly related to the
strength of the material and to the depth of the material removal zone. In
general, the cavity stability depends on a combination of material strength
(UCS,
Maximum compressive stress for representative models after the
same amount of material removal (
The gravitational stress field in the models also means that the absolute
depth, and not just relative depth as expressed by
The DEM models also show how the interaction of material removal and
mechanical instability can lead to cavity growth. This is seen mainly in
moderately strong DEM material (here the “sand and gravel”), where void
spaces usually stay stable until large volumes of material are removed, with
typical spalling at the sides rather than from the roof
(Fig. 8). This lateral spalling of the cavity is
typical of “tunnel breakouts” encountered by engineers and arises from the
in situ stress field in the DEM model surrounding the cavity being
characterised by a
Another important result of our DEM models is that multilayer models with a
weak (mud) interlayer fail earlier than the models with a uniform material.
This is not only because the integrated strength of the overburden is
lessened, but also because the rapid failure of any cavities in the weak
layer effectively increases the stress concentration in the strong overlying
layer, similar to a beam (Fig. 16), leading to
bending induced stresses with inner arc contraction and outer arc extension.
This is contrary to the higher
A consequence of such material-controlled cavity stability is that, as is often inferred for nature (e.g. Waltham et al., 2005), the geometric relationship between subsurface cavities and sinkholes is not a straightforward one. In the weak DEM model material, a sinkhole can have little or no geometric relationship to a cavity, because cavities are not sustained at any comparable scale. In the strong DEM model materials, on the other hand, the sinkhole geometry may relate to cavity geometry to a variable degree. This relationship may be especially direct in the case of a shallow removal zone and a very strong material, where a cavity can stably grow upward with little or no collapse until intersecting the ground surface. Overall, our results reinforce the point that the use of continuum-based methods to estimate cavity geometry from sinkhole geometry (i.e. where there are large permanent strains) should be treated with caution (see also Fuenkajorn and Archeeploha, 2010 and Holohan et al., 2017).
Future work will include a variation of lateral (long-wall-mining-like), vertical (tube-like) and multiple void space growth systems. Especially for typical karst simulations, multiple void spaces with different growth functions and geometries are a more suitable, complex approach. Another aspect is the role of hydrostatic (buoyancy) and pore pressure, which is usually an important factor regarding soil liquefaction and landslides due to the reduction of effective stress (cf., e.g. Tharp, 1999; Zeev et al., 2017; Clément et al., 2018) and has been ignored in these simulations for simplicity. A possible DEM approach is to apply forces to the boundary particles of the void space to simulate the pressure inside a water-filled cavity or to apply forces related to the pore spaces between particles to simulate hydrofractures (Yoon et al., 2015). An alternative is the combination of FEM and DEM with accounting for drag forces due to fluid flow or other combined particle-lattice model schemes (Ghani et al., 2013).
In general, the good fit of model sinkhole geometry with the observed
topography of sinkholes at Ghor Al-Haditha (Sect. 4) confirms the suitability of the DEM approach and
allows for interpretation of morphological features there. In addition,
structures as found in the simulations are visible also in the field, such
as sagging layers and distributed marginal fracturing in weak materials, as
well as cavities, compression ridges (pop-up structures) and overhanging
sides in stronger materials. For a still better fit to the low diameter
results of the field (Fig. 15), we would need to
use a wider variety of the void space growth functions, geometries and
subrosion zone depths, as expected to happen in nature. Due to computational
costs, this has not been included in this study. Nonetheless, the already
good agreement between the paths of depth
Simulated sinkhole depth
Since material heterogeneity is the rule rather than the exception in nature, and since our simulation results fit well to seismic and photogrammetric studies in the area of Ghor Al-Haditha (Al-Halbouni et al., 2017; Polom et al., 2018), we consider our multilayer models as favourable over uniform models for Ghor Al-Haditha. The exact values of large-scale material strength, however, due to the described material testing procedure with a constant particle packing porosity and the limitations of literature laboratory scale values under the assumption of intact rock, should be rather used carefully. Lower strength for the materials in the field is highly probable, as the observed maximum crack depth in alluvial and salt materials (4 m) is less than in the DEM simulations (up to 12 m). This is probably because in detail “pure” sand–gravel or rock salt is rare on a large scale at the site – usually, there is plenty of muddy material interbedded. However, some general observations for the models with materials and material successions typical at the field site of Ghor Al-Haditha in Jordan can be drawn from the following simulations:
A weak lacustrine mud layer beneath a strong cover material favours sinkhole
formation. Even high-strength material like the salt would collapse in such
a setting. A middle–deep subrosion zone (30–40 m) leads to collapses even for the
pure alluvium models, which means that a subrosion acting only in the
alluvial sediments can similarly cause sinkhole formations like those with a
weak interlayer. Only a higher-volume removal is needed. The pure salt models do not produce typical sinkholes as observed in the
field zone. This fact can be related either to a lack of such a thick and
strong cover material in nature or a too-high strength assigned in the
model. It is perhaps worth noting that at the Lisan Peninsula, close to the
field area at the Dead Sea, large (several-metre-scale) cavities and arches
were observed here in Holocene Dead Sea salt (Closson et
al., 2007). On the other hand, the observed salt exposure at our field site
contains rather thin salt layers, interleaved with mud on a centimetre scale, so that
the bulk material strength there is expected to be lower than that
simulated. The possibility to record surface subsidence before actual sinkhole collapse
depends on both the cover material type and the depth of the subrosion zone.
A multilayer model of a middle-deep subrosion zone with a large subsurface
collapse zone may produce recordable surface signatures in the order of sub
cm before the onset of collapse.
Finally, the single void collapse concept explored in this paper may sufficiently explain some individual sinkhole occurrences at Ghor Al-Haditha and elsewhere around the Dead Sea (cf. laboratory experiments by Oz et al., 2016); the coalescence, sequence evolution and sinkhole cluster structures, morphological expressions at the surface and larger sinkhole depression areas may not. For this, a more sophisticated approach of multiple void space growth, testing different geometries and a more realistic subrosion process is necessary and will be addressed in a future paper.
In this work, we presented a benchmarked and calibrated 2-D distinct element modelling approach to simulating the process of both cavity growth and sinkhole development. Our principal findings are as follows.
Firstly, we presented a computationally fast DEM approach to simulating sinkhole formation by instantaneous, quasi-static, stepwise material removal in a single void space at a depth of an arbitrarily shaped geometry under gravitational loading. We successfully benchmarked the models with analytical and BEM solutions yielding a sub-millimetre degree of agreement for surface displacements and displacement differences.
Secondly, we performed simulated compression and tension tests to determine
microscopic bond strength parameters and moduli calibrated by intact rock
literature values and field estimates for the three materials common at the
Dead Sea shoreline. The simulated rock tests yield low bulk strength (UCS
Thirdly, we simulated a cavity growth until sinkhole collapse in uniform materials. Cavity development is controlled by the interaction of the material strength and the depth of material removal. Weak materials do not support large cavities, and so subsidence is characterised by gradual sagging and suffusion-type collapse into the material removal zone. Stronger materials support the development of large cavities at the material removal zone, leading to sinkhole formation by the sudden collapse of the overburden (caprock or cover collapse type sinkholes). At one end of the spectrum, near the Earth's surface, very strong materials may support cavity growth until the intersection with the ground surface, giving rise to sinkholes with little or no collapse. At the other end of the spectrum, below sufficient depth and for a given material strength, the development of cavities on a significant scale is inhibited as gravitational stresses are too high.
Fourthly, we simulated a cavity growth until the sinkhole collapse in multilayered materials. We show with the inclusion of weak layers, either as cover material or as subroded bedrock material, results in sinkhole development with less volume of removed material than in the case of uniform model material. Such development is not only due to an integrated weakening of the overburden but also due to the growth of a subsurface collapse zone in the weak material that geometrically destabilises the overburden.
Lastly, we compare the developed morphologies from a set of models for all
three materials with photogrammetric analysis from the sinkhole area of Ghor
Al-Haditha in Jordan. Our approach produces physically realistic sinkhole
shapes and successfully reproduces typically measured sinkhole depth
The DEM is a specific scheme of undeformable particles and deformable contacts developed by (Cundall, 1971). In the PFC2D v5.035 software, the DEM is used to implement Newton–Euler equations of motion and rotation on disk-shaped particles (Itasca Cooperation Group, 2014; Potyondy, 2014a; Potyondy and Cundall, 2004) (Fig. A1a). In PFC, the resolution scheme is an explicit second-order velocity Verlet algorithm (Verlet, 1967). The particles are assigned a mass and a radius, are initially unbonded and are free to move and rotate depending on external forces. Particles interact only at contact points between particles and wall facets, where the mechanical interaction is treated in terms of a frictional contact with a set of linear elastic springs that are assigned normal and shear stiffness (Fig. A1b). The “rigidity” of the particles is defined by setting the elastic Young constant in accordance to the spring stiffness. An additional bonding of the elements can be performed, whereby many different bond types can be specified. Here, we use the parallel-bond model (Potyondy and Cundall, 2004), which is defined in terms of a set of linear elastic springs in parallel to the linear contact bond. The parallel bonds allow for tensile forces and bending moments between the bonded particles, and they break once their strength is exceeded. Here, we set the bonds to have the same material constants (microproperties) as the particles, like stiffness and elastic modulus, but since bond strength is defined similar to a Mohr–Coulomb failure criterion, the bonds are also assigned a cohesion, tensile strength and friction angle (Fig. A1c).
Schematic description of 2-D DEM modelling with PFC2D v5.
The Newton–Euler equations are solved in a finite difference explicit
time-stepping algorithm involving dynamic relaxation
(Cundall, 1971; Jing and Stephansson, 2007). During the
procedure, Newton's second law and the force-displacement law is solved for
each of the particles and its contacts (Potyondy and
Cundall, 2004). For a 2-D system of coupled rigid elements, the differential
equations solved by the explicit time-marching relaxation scheme for a
particle of mass
It is assumed that (1) velocities and accelerations within one time step are
constant and (2) that the step chosen is small enough that disturbances,
which occur due to external or body forces, particle or boundary wall
movement, propagate only to the neighbours of the particles. The resulting
velocity and acceleration components for both the translational and
rotational motion of one particle are determined via a finite difference
scheme successively for each time step
The equilibrium is defined by a convergence criterion, where the ratio
between the “out-of-balance” forces to the overall forces is below a
defined threshold (solve ratio, SR) of usually
Creation of a bonded particle assembly in this study followed that of Holohan et al. (2011) and involved the following chain of steps:
At each step, the material assembly is cycled until a static equilibrium is
reached. The behaviour of a DEM model depends strongly on the material
packing assembly (Schöpfer et al., 2009), and so a
spectrum of solutions is usually obtained by performing multiple
realisations for different assemblies. The above chain is thus repeated to
produce many random particle assemblies that may be used to obtain a
statistical mean of packing-dependent model outcomes. In this study, the
procedure was repeated generally for 5–10 random assemblies of the
particles.
The following section gives an overview over the performed DEM model convergence, void space installation and benchmarking tests that were performed to determine the optimal sinkhole formation modelling setup. Table B1 summarises the main DEM model parameters used for the tests.
Dimensions of the model and contact and particle properties used in development and testing of DEM cavity formation models.
Several methods have been tested in order to determine the optimal void
installation procedure for reasonable simulation time and realistic surface
displacement curves. These are instantaneous material removal by particle
deletion (M1), incremental material removal by particle deletion (M2), whole
cavity particle radii reduction (M3) and incremental particle radii
reduction (M4). The radius (
M3 and M4 show similar results for the horizontal displacement
Displacement plots for different void installation methods.
We performed model resolution tests to determine the optimal size for the
mechanical problem of a shallow cavity in a bonded rock assembly. The cavity
is installed by instantaneous (quasi-static) particle removal (M1 as shown
in Fig. 4a). We varied the width
In Fig. B2, we see the horizontal and vertical
displacement curves for all model dimensions. Boundary effects in such a
setting close to the free surface make the judgement of the optimal size
demanding, but the expected behaviour for the vertical displacement is a
subsidence roughly
Convergence test results for model assembly dimensions: cavity of
radius of 5 m and depth of 35 m created in each case by method M1. Mean
particle radius is 0.74 m. Left and right columns show horizontal and
vertical displacement profiles, respectively. Each plot shows results for
varying model height (
We observe the most stable results for symmetric model dimensions and define
the optimal model size to height (
The influence of the particle radii on the displacement curves is shown in
Fig. B3 for the above-determined favourable model
dimensions. A convergence is observed for particles with mean radius around
0.32 m. Model dimensions of
Results of convergence tests of for influence of particle size:
horizontal
The first analytical solution used, the Kirsch solution, a classical
solution for simple excavation shapes, does not include the free-surface
effect, and the mathematical details are depicted, e.g. in Brady
and Brown (2006). The radial and tangential displacements at a point
The equation for normal displacements as derived by the second solution for
an elastic half space (
As known well from linear elastic material theory
(Muskhelishvili, 2013; Timoshenko and Goodier, 1973), the
integration of the stress formulae is such that a setting of a loaded material
(Flamant's problem), which is similar to material removal in the
underground, leads to the logarithmic term in the equation above. This leads
to infinite vertical displacements along the
As a workaround for calculation of finite displacements around the cavity,
Verruijt and Booker (2009) defined a value
Thus, displacements are considered as not physically realistic in the
far field of a load (or cavity), but relative displacement differences are
(cf. Davis and Selvadurai, 1996 and Verruijt and Booker, 2009). For the above-stated
problem, the relative vertical displacements
The effect of Young's modulus
The bulk behaviour of particle assemblies emerges from the interaction of
the particle according to the mechanical rules imposed at the contact and
bond scale. Therefore, and unlike for continuum-based approaches, the bulk
behaviour in DEM models must be calibrated by simulated rock or soil
mechanics tests (Potyondy and Cundall, 2004). Here,
biaxial compression and tension tests are used to determine the bulk elastic
properties of the medium, i.e. the Poisson ratio
Differential stress vs. vertical strain for CC and DT tests for a
confining pressure of 0.1 MPa:
A typical stress vs. strain curve contains three parts: (1) a non-linear or
linear elastic behaviour, (2) a non-linear yielding behaviour as cracks
appear in the material and (3) a non-linear post-peak behaviour after
material failure. The peak of the stress–strain curve defines the maximum
and minimum principal stresses (
The mean peak stresses can be determined for each confining pressure and
plotted against each other. In a linear (Mohr–Coulomb) fit of
A graphical description of the implemented Python/PFC2D-Fish sinkhole
modelling code is depicted in Fig. B6. Here, Fish
code parts are marked in yellow and Python code in grey colour. A typical
sinkhole simulation follows the following scheme:
Model dimensions, particle parameters and a function Similar to the material generation procedure for the model verification
material (see Appendix A2), we settle and bond the assembly
with a parallel bond model according to the desired material properties. It has to be noted
that for low-strength material a bond-reinstallation procedure has been
applied, a so-called annealing; i.e. failed bonds can be re-established by
contact with other particles of the same material, accounting for, e.g.
cohesive mud behaviour. For the other materials, a failed PB is not activated
again. We then install the desired tracking functions (measurement circles,
marker particles, histories) and group the initial void spaces defined in
the model control file. This material removal loop acts on each defined cavity growth round Step 3 is repeating with increasing material removal round
Graphical description of the PFC2D-based sinkhole modelling code.
Yellow colours indicate PFC-Fish language-based code; grey colours are
Python control connections. Solid arrows indicate time step cycling. Each
model set consists of
To avoid another degree of freedom in the calibration of micro- vs. macroproperties, the initial porosity only changes due to the compression by the gravity settling scheme. We have refrained from using either post-settling particle removal to adjust the porosities to specific values or layer-wise gravity deposition with different porosities because of the high amount of calculation time needed.
The Fish material removal core loop (no. 3 in Fig. B6) provides the technical implementation of a quasi-static void space
growth. A simple law between the particle area
The pure runtime for a full simulation of an alluvium on mud setup on an
Intel Xeon 3.7 GHz processor with 64 GB RAM needs roughly 2 weeks for one particle
assembly without tracking geophysical parameters. The tracking would
increase the runtime by a factor of
A tracking of pre-, syn- and post-collapse geodetic and geophysical parameters has been implemented in the modelling code (no. 4 in Fig. B6). The technical details are listed as follows.
Porosity, stress and strain rate are recorded using the distribution of
so-called measurement circles of area
Porosity is calculated via
The average stress tensor is calculated in static conditions via
The strain rate tensor
The strain tensor in the measurement region is then calculated by multiplying strain rate components with the simulation time step and summing over the desired period.
Alternatively, strain is calculated via simulated extensometers. For these, pairs of particles which lie either horizontally or vertically next to each other are defined. By registering the displacement of each particle, a pairwise calculation of the horizontal and vertical strains is achieved at low computational cost in comparison to the measurement circle distribution (Itasca Cooperation Group, 2014).
A full set of metadata is available upon request. For photogrammetric surveys, raw images, DSMs and orthophotos are available upon consultation with the authors. For DEM models, data and results are available upon request.
The supplement related to this article is available online at:
DAH led the production of figures, data analysis and numerical modelling. DAH and EPH led the writing of the manuscript. DAH, EPH, AT, MPJS and SE developed the calibration of the materials and discussed technical issues. All authors reviewed and commented on the manuscript, and they contributed to discussions of the data and interpretation.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Environmental changes and hazards in the Dead Sea region (NHESS/ACP/HESS/SE inter-journal SI)”. It is not associated with a conference.
We would like to thank the two reviewers and the editor for the fruitful discussion. We very gratefully acknowledge fieldwork support from Ali Sawarieh, Hussam Alrshdan and their colleagues at the Ministry of Energy and Mineral Resources of the Hashemite Kingdom of Jordan. Acknowledgements go to Emad Talafheh and Zayad Mansour from the Arab Potash Company, as well as to Michael Ezersky of the Geophysical Institute of Israel, for their support regarding geotechnical parameters. Also, we thank Arnold Verruijt for the comments regarding the analytical solution, as well as Daniel Woodell and Mehdi Nikhoo for the technical advice of the DEM and BEM modelling, respectively, and Marc Elmouttie for facilitating the research stay at CSIRO (Brisbane). Special thanks go to Damien Closson and Thomas R. Walter for continuous support and material provision. Particular thanks go to Itasca Consulting Group for providing the license of PFC2D v5.0 in the framework of the Itasca Education Partnership program and the German Academic Exchange Service (DAAD) for a short-term doctorate research grant. Last, but not least, thanks go to the projects DESERVE and SIMULTAN and the involved colleagues for their kind support and funding opportunities. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Amotz Agnon Reviewed by: Renaud Toussaint and one anonymous referee